Surface Waves in a Collisional Quark-Gluon Plasma

Abstract

Surface waves propagating in the semi-bounded collisional hot QCD medium (quark-gluon plasma) are considered. To investigate the effect of collisions as damping and non-ideality factor, the longitudinal and transverse dielectric functions of the quark-gluon plasma are used within the Bhatnagar–Gross–Krook (BGK) approach. The results were obtained both analytically and numerically in the long wavelength limit. First of all, collisions lead to smaller values of surface wave frequency and their stronger damping. Secondly, the results show that non-ideality leads to the appearance of a new branch of surface waves compared to the collisionless case. The relevance of the surface excitations (waves) for the QGP realized in experiments is discussed.

APPENDIX A

Taking into account that $a \ll 1$, we used the following expansion:

$$ln\frac{{1 + a}}{{1 - a}} \approx 2a + \frac{{2{{a}^{3}}}}{3} + \frac{{2{{a}^{5}}}}{5} + \ldots $$

(A.1)

Substituting the expansion (A.1) into Eqs. (5) and (6) we get

$$\begin{gathered} {{\epsilon }_{l}}(\tilde {\omega },\tilde {k}) \approx 1 + \frac{1}{{{{{\tilde {k}}}^{2}}}}\left( { - \frac{{{{a}^{2}}}}{3} - \frac{{{{a}^{4}}}}{5}} \right) \\ \times \,\,{{\left( {1 - \frac{{\widetilde {i\tilde {\nu }}}}{{\tilde {k}}}a - \frac{{i\tilde {\nu }}}{{3\tilde {k}}}{{a}^{3}} - \frac{{i\tilde {\nu }}}{{5\tilde {k}}}{{a}^{5}}} \right)}^{{ - 1}}}, \\ \end{gathered} $$

(A.2)

and

$${{\epsilon }_{t}}(\tilde {\omega },\tilde {k}) \approx 1 - \frac{a}{{2\tilde {\omega }\tilde {k}}}\left\{ {1 + \left( {\frac{1}{{{{a}^{2}}}} - 1} \right)\left( { - \frac{{{{a}^{2}}}}{3} - \frac{{{{a}^{4}}}}{5}} \right)} \right\},$$

(A.3)

respectively. Neglecting terms of $\mathcal{O}({{a}^{3}})$ in Eqs. (A.2) and (A.3), and making use of the expansion ${{\left( {1 - \tfrac{{i\tilde {\nu }}}{{\tilde {k}}}a} \right)}^{{ - 1}}} \approx 1 + \tfrac{{i\tilde {\nu }}}{{\tilde {k}}}a$ in Eq. (A.2), we arrive at

$$\begin{gathered} {{\epsilon }_{l}}(\tilde {\omega },\tilde {k}) \approx 1 + \frac{1}{{{{{\tilde {k}}}^{2}}}}\left( { - \frac{{{{a}^{2}}}}{3}} \right){{\left( {1 - \frac{{i\tilde {\nu }}}{{\tilde {k}}}a} \right)}^{{ - 1}}} \\ \approx 1 + \frac{1}{{{{{\tilde {k}}}^{2}}}}\left( { - \frac{{{{a}^{2}}}}{3}} \right)\left( {1 + \frac{{i\tilde {\nu }}}{{\tilde {k}}}a} \right), \\ \end{gathered} $$

(A.4)

and

$$\begin{gathered} {{\epsilon }_{t}}(\tilde {\omega },\tilde {k}) \approx 1 - \frac{a}{{2\tilde {\omega }\tilde {k}}}\left\{ {1 + \left( {\frac{1}{{{{a}^{2}}}} - 1} \right)\left( { - \frac{{{{a}^{2}}}}{3} - \frac{{{{a}^{4}}}}{5}} \right)} \right\} \\ = 1 - \frac{a}{{2\tilde {\omega }\tilde {k}}}\left\{ {\frac{2}{3} + \frac{{2{{a}^{2}}}}{{15}}} \right\}, \\ \end{gathered} $$

(A.5)

Finally, dropping terms of $\mathcal{O}({{a}^{3}})$ in Eqs. (A.4) and (A.5), we get expressions (7) and (8).